By combining the DiPerna and Lions techniques for the nonrelativistic Boltzmann equation and the Dudyński and Ekiel-Jeżewska device of the causality of the relativistic Boltzmann equation, it is shown that there exists a global mild solution to the Cauchy problem for the relativistic Boltzmann equation with the assumptions of the relativistic scattering cross section including some relativistic...

to start with, having employed transformation wave, some nonlinear partial differential equations have been converted into an ode. then, using the infinite series method for equations with similar linear part, the researchers have earned the exact soliton solutions of the selected equations. it is required to state that the infinite series method is a well-organized method for obtaining exact s...

Algebro-geometric Poisson brackets for real, finite-zone solutions of the Korteweg–de Vries (KdV) equation were studied in [1]. The transfer of this theory to the Toda lattice and the sinh-Gordon equation is more or less obvious. The complex part of the finite-zone theory for the nonlinear Schrödinger equation (NS) and the sine-Gordon equation (SG) is analogous to KdV, but conditions that solut...

We apply the Painlevé test for integrability to the new discrete nonlinear Schrödinger equation introduced by Leon and Manna. Since the singular expansions of solutions of this equation turn out to contain non-dominant logarithmic terms, we conclude that the studied equation is nonintegrable. This result supports the recent observation of Levi and Yamilov that the Leon–Manna equation does not a...

In a recent breakthrough Kanzieper showed that it is possible to obtain exact nonperturbative random matrix results from the replica limit of the corresponding Painlevé equation. In this article we analyze the replica limit of the Toda lattice equation and obtain exact expressions for the two-point function of the Gaussian unitary ensemble and the resolvent of the chiral unitary ensemble. In th...

A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differentialdifference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classic...

Bilinear approach is applied to derive integrable multi-component generalizations of the socalled 1+1 dimensional special Toda lattice, the Volterra lattice, a simple differential-difference equation found by Adler, Moser, Weiss, Veselov and Shabat and another integrable lattice reduced from the discrete BKP equation. Their soliton solutions expressed by pfaffians and the corresponding bilinear...

We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differentialdifference equations. As an example of the application of the methods, we construct the discrete symmetries of the discrete Painlevé I equation and of the Toda lattice equation.

We present a nonlinear theory for relativistic x-ray free-electron lasers in the quantum regime, using a collective Klein-Gordon (KG) equation (for relativistic electrons), which is coupled with the Maxwell-Poisson equations for the electromagnetic and electrostatic fields. In our model, an intense electromagnetic wave is used as a wiggler which interacts with a relativistic electron beam to pr...